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Computer Science > Computational Complexity

Abstract: In this paper we give subexponential size hitting sets for bounded depth
multilinear arithmetic formulas. Using the known relation between black-box PIT
and lower bounds we obtain lower bounds for these models.
For depth-3 multilinear formulas, of size $\exp(n^\delta)$, we give a hitting
set of size $\exp(\tilde{O}(n^{2/3 + 2\delta/3}))$. This implies a lower bound
of $\exp(\tilde{\Omega}(n^{1/2}))$ for depth-3 multilinear formulas, for some
explicit polynomial.
For depth-4 multilinear formulas, of size $\exp(n^\delta)$, we give a hitting
set of size $\exp(\tilde{O}(n^{2/3 + 4\delta/3}))$. This implies a lower bound
of $\exp(\tilde{\Omega}(n^{1/4}))$ for depth-4 multilinear formulas, for some
explicit polynomial.
A regular formula consists of alternating layers of $+,\times$ gates, where
all gates at layer $i$ have the same fan-in. We give a hitting set of size
(roughly) $\exp\left(n^{1- \delta} \right)$, for regular depth-$d$ multilinear
formulas of size $\exp(n^\delta)$, where $\delta = O(\frac{1}{\sqrt{5}^d})$.
This result implies a lower bound of roughly
$\exp(\tilde{\Omega}(n^{\frac{1}{\sqrt{5}^d}}))$ for such formulas.
We note that better lower bounds are known for these models, but also that
none of these bounds was achieved via construction of a hitting set. Moreover,
no lower bound that implies such PIT results, even in the white-box model, is
currently known.
Our results are combinatorial in nature and rely on reducing the underlying
formula, first to a depth-4 formula, and then to a read-once algebraic
branching program (from depth-3 formulas we go straight to read-once algebraic
branching programs).